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A137346 Coefficients of a special case of Poisson-Charlier polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n. 4
1, -2, 1, 4, -5, 1, -8, 20, -9, 1, 16, -78, 59, -14, 1, -32, 324, -360, 135, -20, 1, 64, -1520, 2254, -1165, 265, -27, 1, -128, 8336, -15232, 9954, -3045, 469, -35, 1, 256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1, -512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..53.

M. Dunster, Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001) pp. 93-133.

Eric Weisstein's World of Mathematics, Poisson-Charlier Polynomial.

FORMULA

T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(-2*t)*(1+t)^x.

With p(0, x) = 1 and p(1, x) = x - 2 the polynomials obey the recurrence

p(n, x) = (x - n - 1)*p(n-1, x) - 2*(n - 1)*p(n-2, x).

Row sums are (-2)^n*(n-1) = (-1)^n*A159964(n-1).

From Peter Bala, Oct 23 2019: (Start)

The unsigned row polynomials are

   R(n,x) = Sum_{k=0..n} (-1)^k*binomial(n, k)*k!*2^(n-k)*binomial(-x, k).

They occur in series acceleration formulas for the constant

   1/e^2 = n!*2^n*Sum_{k >= 0}(-2)^k/(k!*R(n,k)*R(n,k+1)) = 0.1353 35283 23661 ... (cf. A092553, A046716, A094816).

(End)

R(n, x) = KummerU(-n, 1 - n - x, 2). - Peter Luschny, Oct 27 2019

EXAMPLE

{1},

{-2, 1},

{4, -5, 1},

{-8, 20, -9, 1},

{16, -78,59, -14, 1},

{-32, 324, -360, 135, -20, 1},

{64, -1520, 2254, -1165, 265, -27, 1},

{-128, 8336, -15232, 9954, -3045, 469, -35, 1},

{256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1},

{-512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54, 1},

{1024, -3492416, 8793216, -8592220,4373060, -1297569, 235473, -26370, 1770, -65, 1}

MAPLE

R := proc(n) add((-1)^k*binomial(n, k)* k!*2^(n-k)*binomial(-x, k), k=0..n);

expand(%) end: p := n -> seq((-1)^(n-k)*coeff(R(n), x, k), k=0..n):

seq(p(n), n = 0..9);

# Or:

egf := exp(-2*t)*(1+t)^x: ser := series(egf, t, 12): p := n -> coeff(ser, t, n):

seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # Peter Luschny, Oct 27 2019

MATHEMATICA

Ca[x, 0] = 1; Ca[x, 1] = -2 + x;

Ca[x_, n_] := Ca[x, n] = (x - n - 1) Ca[x, n - 1] - 2 (n - 1) Ca[x, n - 2];

Table[CoefficientList[Ca[x, n], x], {n, 0, 9}] // Flatten

(* The unsigned row polynomials (see Peter Bala's comment) are: *)

R[n_] := HypergeometricU[-n, 1 - n - x, 2];

Table[R[n], {n, 0, 6}] (* Peter Luschny, Oct 27 2019 *)

CROSSREFS

Cf. A046716, A092553, A094816, A159964.

Sequence in context: A154342 A143494 A124960 * A264017 A159971 A114158

Adjacent sequences:  A137343 A137344 A137345 * A137347 A137348 A137349

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Apr 08 2008

EXTENSIONS

Edited by Peter Luschny, Oct 27 2019

STATUS

approved

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Last modified August 2 15:51 EDT 2021. Contains 346428 sequences. (Running on oeis4.)